OFFICIAL ORGAN OF THE RADIATION RESEARCH SOCIETY

RADIATION RESEARCH

EDITOR-IN-CHIEF: DANIEL BILLEN

Vo lume 9 1 , ' 1 9 8 2

ACADEMIC P R E S S New York London

Paris San Diego San Francisco Säo Paulo Sydney Tokyo Toronto

B l B L l p T H ^ C R Tie r s:- 21! \ h*n<r,.: < u 1 <••-1

11, Universitäts-Bibifofhek München

Copyright © 1982 by Academic Press, Inc.

All rights reserved

No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the copyright owner.

The appearance of the code at the bottom of the first page of an article in this journal indicates the copyright owner's consent that copies of the article may be made for personal or internal use, or for the personal or internal use of specific clients. This consent is given on the condition, however, that the copier pay the stated per copy fee through the Copyright Clearance Center, Inc. (21 Congress Street, Salem, Massachusetts 01970), for copying beyond that permitted by Sections 107 or 108 of the U. S. Copyright Law. This consent does not extend to other kinds of copying, such as copying for general distribution, for advertising or promotional purposes, for creating new collective works, or for resale. Copy fees for pre-1982 articles are the same as those shown for current articles.

MADE IN THE UNITED STATES OF AMERICA

R A D I A T I O N KüütiAKLTi

OFFICIAL ORGAN OF THE RADIATION RESEARCH SOCIETY

Editor-in-Chief: DANIEL BILLEN, University of Tennessee-Oak Ridge Graduate School of Biomedical Sciences, Biology Division, Oak Ridge National Laboratory, P.O. Box Y,

Oak Ridge, Tennessee 37830

Managing Technical Editor: MARTHA EDINGTON, University of Tennessee-Oak Ridge Graduate School of Biomedical Sciences, Biology Division, Oak Ridge National Laboratory, P.O. Box Y,

Oak Ridge, Tennessee 37830

ASSOCIATE EDITORS

H. I. ADLER, Oak Ridge National Laboratory R. E. MEYN, JR., University of Texas

J. W. BAUM, Brookhaven National Laboratory S. M. MICHAELSON, University of Rochester

W. D. BLOOMER, Harvard Medical School S. OKADA, University of Tokyo, Japan

S. S. BOGGS, University of Pittsburgh M. SODICOFF, Temple University

A. COLE, University of Texas J. R. STEWART, University of Utah

L. L. DEAVEN, Los Alamos National Labo- R. C. THOMPSON, Battelle, Pacific Northwest ratory Laboratories

S. S. DONALDSON, Stanford University J. E. TURNER, Oak Ridge National Laboratory

J. J. FISCHER, Yale University S. S. WALLACE, New York Medical College

E. W. GERNER, University of Arizona J. F. WARD, University of California, San Diego

A. HAN, Argonne National Laboratory D. W. WHILLANS, Ontario Hydro, Pickering, Canada

OFFICERS OF THE SOCIETY

President: H. RODNEY WITHERS, Department of Radiation Oncology, UCLA Center for Health Sciences, Los Angeles, California 90025

Vice President and President-Elect: EDWARD R. EPP, Department of Radiation Medicine, Massachusetts General Hospital, Boston, Massachusetts 02114

Secretary-Treasurer: ROGER O. McCLELLAN, Inhalation Toxicology Research Institute, The Lovelace Foundation, P.O. Box 5890, Albuquerque, New Mexico 87115

Editor-in-Chief: DANIEL BILLEN, University of Tennessee-Oak Ridge Graduate School of Biomedical Sciences, Biology Division, Oak Ridge National Laboratory,

P.O. Box Y, Oak Ridge, Tennessee 37830 Administrative Director: JOHN J. CURRY, 925 Chestnut Street, Philadelphia, Pennsylvania 19107

ANNUAL MEETINGS

1983: February 27-March 3, San Antonio, Texas 1984: April 8-12, Orlando, Florida 1985: May 26-31, Chicago, Illinois

Titus C. Evans, Editor-in-Chief Volumes 1-50 Oddvar F. Nygaard, Editor-in-Chief Volumes 5 1 - 7 9

VOLUME 9 1 , 1982

Councilors, Radiation Research Society 1981-1982

P H Y S I C S

R. M . Henkelman, Ontario Cancer Institute, Toronto, Canada L. D. Skarsgard, British Columbia Cancer Research Center,

Vancouver, Canada

B I O L O G Y

A . M . Rauth, Ontario Cancer Institute, Toronto, Canada J. M . Brown, Stanford University

M E D I C I N E

J. A . Belli, Harvard Medical School S. S. Donaldson, Stanford University

C H E M I S T R Y

M . Z. Hoffman, Boston University J. D. Zimbrick, University of Kansas

A T - L A R G E

L. A . Dethlefsen, University of Utah R. M . Sutherland, University of Rochester

CONTENTS OF VOLUME 91

NUMBER 1, J U L Y 1982

EDITORIAL iü

PLENARY SESSIONS: NUCLEAR WASTE MANAGEMENT

RICHARD K. L E S T E R . Nuclear Waste Management: Some Considerations of Scale 1 F R E D A. DONATH. The Isolation of High-Level Nuclear Wastes in a Geologic Repository 22 DANIEL M E T L A Y . The Institutional Aspects of Radioactive Waste Management 34

REVIEW

D U D L E Y T. GOODHEAD. An Assessment of the Role of Microdosimetry in Radiobiology .. 45

REGULAR A R T I C L E S

TOORU KOBAYASHI AND K E I J I KANDA. Analytical Calculation of Boron-10 Dosage in Cell Nucleus for Neutron Capture Therapy 77

M. ZAIDER, D. J. BRENNER, K. HANSON, AND G. N. MINERBO. An Algorithm for Deter­mining the Proximity Distribution from Dose-Averaged Lineal Energies 95

Y V O N N E C. T A Y L O R AND ANDREW M. RAUTH. Oxygen Tension, Cellular Respiration, and Redox State as Variables Influencing the Cytotoxicity of the Radiosensitizer Misoni-dazole 104

R. VAN BRUWAENE, G. B. G E R B E R , R. KIRCHMANN, J. VAN DEN HOEK, AND J . VANKERKOM. Tritium Metabolism in Young Pigs after Exposure of the Mothers to Tritium Oxide during Pregnancy 124

V. V. Y A N G AND E. J . AINSWORTH. Late Effects of Heavy Charged Particles on the Fine Structure of the Mouse Coronary Artery 135

G. H. THOMAS, M. A. MALONEY, AND J . E. C L E A V E R . Sensitization of Mouse L Cells to Ultraviolet Light by Low Amounts of Bromodeoxyuridine 145

B R U C E F. K I M L E R AND SHERI D. HENDERSON. Cyclic Responses of Cultured 9L Cells to Radiation 155

DOUGLAS E. BRASH AND RONALD W. HART. Biopsy Measurement of DNA Damage and Repair in Vivo: Single-Strand Breaks, Alkali-Labile Bonds, and Endonuclease-Sensitive Sites 169

BASIL V. WORGUL, SUJON LOW, AND G E O R G E R. M E R R I A M , JR. The Lens Epithelium and Radiation Cataract. III. The Influence of Age on the Nuclear Fragmentation of the Meridional Row Cells following X Irradiation 181

H E L E N B. STONE AND MARK S. SINESI. Testing of New Hypoxic Cell Sensitizers in Vivo 186 K A R E N L. BEETHAM AND L. J . TOLMACH. The Action of Caffeine on X-Irradiated HeLa

Cells. V. Identity of the Sector of Cells That Expresses Potentially Lethal Damage in G, and G 2 199

DONALD I. M C R E E AND HOWARD WACHTEL. Pulse Microwave Effects on Nerve Vitality 212

NUMBER 2, AUGUST 1982

M. M O L L S , C. STREFFER, D. VAN BEUNINGEN, AND N. ZAMBOGLOU. X Irradiation in G 2 Phase of Two-Cell Mouse Embryos in Vitro: Cleavage, Blastulation, Cell Kinetics, and Fetal De­velopment 219

PETER A. MAHLER, M I C H A E L N. GOULD, PAUL M. D E L U C A , DAVID W. PEARSON, AND K E L L Y H. CLIFTON. Rat Mammary Cell Survival following Irradiation with 14.3-MeV Neutrons 235

HIROO KATO, C H A R L E S C. BROWN, DAVID G. H O E L , AND W I L L I A M J . S C H U L L . Studies of the Mortality of A-Bomb Survivors 243

K E N N E T H L. MOSSMAN. Quantitative Radiation Dose-Response Relationships for Normal Tissues in Man. I. Gustatory Tissue Response during Photon and Neutron Radiotherapy 265

ABSTRACTS OF PAPERS FOR T H E THIRTIETH ANNUAL M E E T I N G OF T H E RADIATION R E S E A R C H S O C I E T Y 275

ABSTRACTS OF PAPERS FOR T H E ANNUAL M E E T I N G OF T H E NORTH AMERICAN HYPERTHERMIA

GROUP 414

ANNOUNCEMENTS 429

OBITUARY 431

NUMBER 3, SEPTEMBER 1982

H. JUNG. Interaction of Thermotolerance and Thermosensitization Induced in CHO Cells by Combined Hyperthermic Treatments at 40 and 43°C 433

A K I K O M. UENO, IKUKO FURUNO-FUKUSHI, AND HIROMICHI MATSUDAIRA. Induction of Cell Killing, Micronuclei, and Mutation to 6-Thioguanine Resistance after Exposure to Low-Dose-Rate y Rays and Tritiated Water in Cultured Mammalian Cells (L5178Y) 447

D. BETTEGA, A. M. FUHRMAN CONTI, L. GARIBOLDI, M. T. PELUCCHI , E. SCAIOLI, AND L. T A L L O N E LOMBARDI. Age Response of E U E Cells Exposed to 31-MeV Protons 457

O L E S . N I E L S E N , K U R T J . H E N L E , AND J E N S OVERGAARD. Arrhenius Analysis of Survival Curves from Thermotolerant and Step-Down Heated L1A2 Cells in Vitro 468

G E O R G E ILIAKIS AND M I C H A E L NUSSE. Conditions Supporting Repair of Potentially Lethal Damage Cause a Significant Reduction of Ultraviolet Light-Induced Division Delay in Syn­chronized and Plateau-Phase Ehrlich Ascites Tumor Cells 483

JOSEF K R I Z A L A , A L E N A STOKLASOVA, HANA KOVAROVA, AND MIROSLAV LEDVINA. The Effect of 7 Irradiation and Cystamine on Superoxide Dismutase Activity in the Bone Marrow and Erythrocytes of Rats 507

SONG-MAO C H I U AND NANCY L. OLEINICK. Resistance of the Nucleosomal Organization of Eucaryotic Chromatin to Ionizing Radiation 516

JOHN A. STRAND, M. P A U L FUJIHARA, T E D M. POSTON, A N D C . SCOTT ABERNETHY. Permanence of Suppression of the Primary Immune Response in Rainbow Trout, Salmo gairdneri, Sub-lethally Exposed to Tritiated Water during Embryogenesis 533

J A N E T S. R A S E Y , KENNETH A. KROHN, AND SARA FREAUFF. Bromomisonidazole: Synthesis and Characterization of a New Radiosensitizer 542

STEPHEN P. TOMASOVIC, HOWARD D. THAMES, JR. , AND GARTH L. NICOLSON. Heterogeneity in Hyperthermic Sensitivities of Rat 13762NF Mammary Adenocarcinoma Cell Clones of Differing Metastatic Potentials 555

SHOZO SUZUKI, M I E K O OSHIMA, AND Y U Z U R U AKAMATSU. Radiation Damage to Membranes of the Thermophilic Bacterium, Thermus thermophilus HB-8: Membrane Damage without Concomitant Lipid Peroxidation 564

A N T O N E L. BROOKS, STEPHEN A. BENJAMIN, ROBERT K. JONES, AND ROGER O. M C C L E L L A N . Interaction of 1 4 4 Ce and Partial Hepatectomy in the Production of Liver Neoplasms in the Chinese Hamster 573

D. C H M E L E V S K Y , A. M. K E L L E R E R , J . LAFUMA, AND J . CHAMEAUD. Maximum Likelihood Estimation of the Prevalence of Nonlethal Neoplasms—An Application to Radon-Daughter Inhalation Studies 589

VINCENZO C O V E L L I , VINCENZO DI MAJO, BRUNO BASSANI, PIETRO M E T A L L I , AND GIOVANNI SILINI . Pattern of Leukemia Induction in BC3F, Mice Transplanted with Irradiated Lym-phohemopoietic Tissues 615

MARK BERNSTEIN, PHILIP H. GUTIN, K E I T H A. WEAVER, DENNIS F. D E E N , AND M A R Y H E L E N BARCELLOS. 1 2 5 I Interstitial Implants in the RIF-1 Murine Flank Tumor: An Animal Model for Brachytherapy 624

CORRESPONDENCE

L. WOJNAROVITS AND G. FOLDIAK. Structure Effect in Alkane Radiolysis 638

BOOK REVIEWS

ROBERT G. THOMAS. Actinides in Man and Animals. Edited by M . E. Wrenn 644

MARTHA EDINGTON. Radioactive Decay Data Tables. By David C . Kocher 646

ANNOUNCEMENTS 647

AUTHOR INDEX FOR V O L U M E 91 648

The Subject Index for Volume 91 will appear in the December 1982 issue as part of a cumulative index for the year 1982.

RADIATION R E S E A R C H 91, 589-614 (1982)

Maximum Likelihood Estimation of the Prevalence of Nonlethal Neoplasms—An Application to Radon-Daughter

Inhalation Studies1

D . C H M E L E V S K Y A N D A . M . K E L L E R E R 2

Institut für Medizinische Strahlenkunde der Universität Würzburg, Versbacher Str. 5, D-8700 Würzburg, Germany

J . L A F U M A

Commissariat ä l'Energie Atomique, Service de Radiopathologie et de Toxicologic Experimentale, B.P. 6, F-92260 Fontenay-aux-Roses, France

AND

J . C H A M E A U D

Commissariat ä l'Energie Atomique, Service Medical, B.P. 1, F-87640 Razes, France

CHMELEVSKY, D., K E L L E R E R , A. M., LAFUMA, J. , AND CHAMEAUD, J. Maximum Like­lihood Estimation of the Prevalence of Nonlethal Neoplasms—An Application to Radon-Daughter Inhalation Studies. Radiat. Res. 91, 589-614 (1982).

A nonparametric maximum likelihood method for estimating prevalence is described that is applicable to the analysis of nonlethal tumors which are discovered incidentally in either sacrificed animals or animals dead from other causes. The method corrects for competing risks and does not require an analytical model for the prevalence as a function of dose and time. It is applied to the results of an experiment in which numerous groups of Sprague-Dawley rats were exposed to different doses of radon daughters at different dose rates. The dependence of the prevalence on dose and time after exposure is derived, and three basic models are considered that correspond to a dose-dependent shift in time, to an acceleration in time, and finally to the proportional hazards model. Mortality-corrected risk estimates are derived from the estimated prevalences. At doses down to 65 WLM (working level months) the results are consistent with linearity in dose, or possibly with sublinearity (dose exponent less than 1); they exclude, in this dose range, a threshold or a proportionality to a higher power of dose.

INTRODUCTION

Considerable efforts have been made at a number of laboratories to assess pul­monary tumors due to inhaled radioactive substances. Recent reports (7-5) attest to the broad range of such investigations. They also demonstrate the difficulties

1 Work partly supported by Euratom Contracts 099-76-1-PSA F and 208-76-BIO D. 2 Author to whom correspondence should be addressed.

589 0033-7587/82/090589-26$02.00/0 Copyright © 1982 by Academic Press, Inc. All rights of reproduction in any form reserved.

590 CHMELEVSKY E T AL.

inherent in these studies and emphasize large uncertainties of the risk estimates and the dose-effect relations.

The complexities arise largely from the multitude of different factors involved in these studies. Also an important problem is the lack of mathematical methods to separate the effect of life shortening in the survival experiments from the in­duction of pulmonary tumors. In the analysis of lethal cancers or cancers that can be readily discovered in the live animal, it is common practice to use the proper competing-risk-corrected quantities, for example, the Kaplan-Meier estimate (4). With nonlethal diseases, such as radon-induced pulmonary tumors in Sprague-Dawley rats, the situation is different; dose-effect relations are either reported for life shortening alone or, i f a dose-effect relation is given for the neoplastic process, it is based on incidences uncorrected for life shortening. The failure to utilize a competing-risk-corrected analysis is striking since Hoel and Walburg (5) considered the problem in depth almost a decade ago and pointed out that isotonic regression can take the place of the Kaplan-Meier estimate in the analysis of nonlethal diseases in survival experiments.

The objective of the present article is to demonstrate that the method recom­mended by Hoel and Walburg is applicable to inhalation studies and to show that it can be considerably extended. The method wi l l be exemplified by its application to radon-daughter inhalation studies that are part of the broader range of inhalation investigations at the Commissariat ä l'Energie Atomique (6-8). The separate con­sideration of the radon studies has the advantage that complicating factors such as chemical state or particulate size need not be considered and that, accordingly, the emphasis can be on the essentials of the numerical methods.

MATERIALS AND METHODS AND BACKGROUND OF T H E EXPERIMENTAL WORK

Experimental work on the carcinogenic action of inhaled a emitters began at the Commissariat ä l'Energie Atomique in 1969. Research was performed mainly on male rats of the Sprague-Dawley S.P.F. (specific-pathogen free) and the Wistar A.G. inbred strains. The investigations were directed at two main objectives. The first was the assessment of the effects of the decay products of radon (9, 70), a problem particularly relevant to the mining of uranium; the second was the inves­tigation of effects of the actinides that are of concern in reactor fuel reprocessing plants (77, 72).

When the experimental program began, the carcinogenic effect of radon on rodents was unknown. For this reason doses and exposure times differed widely in the initial experiments. Wi th the intent to reach broader conclusions on basic mechanisms, the studies were subsequently extended to include investigations on the influence of factors such as sex or age at the start of inhalation (7). Experiments were also performed on the combined action of various cofactors which were ex­pected to counteract the neoplastic processes, such as bleomycin, or to enhance i t , such as tobacco smoke, dust, or various chemical compounds (13-16). For the present analysis none of the more complex investigations were utilized.

Even the radon-daughter experiments encompassed a wide range of experiments performed over the last 10 years at the laboratory of experimental pulmonary pathology in Razes that is directed by J. Chameaud (9, 70, 77, 18). The inhalation

ESTIMATION OF T H E PREVALENCE OF NEOPLASMS 591

prqcedure has been described earlier (19). In summary, male Sprague-Dawley rats, 90 days old at the beginning of the experiments, were kept in an inhalation chamber with radon concentration at the desired levels. They were exposed to radon and its daughter products at the concentrations and for the durations specified in each individual experiment. There were 2- to 16-hr exposures two to five times a week. The experimental conditions were such that equilibrium of radon with its short-lived daughters existed. Table I gives parameters for individual experiments. In the first column the total inhalation doses and the monthly doses are listed. In columns 2 to 5 the daily durations of exposure to radon daughters, the weekly frequencies of the exposures, the total durations of the inhalation periods, and the radon-daughter concentrations are given. In columns 6-10 the total number of animals in a group, the number of animals examined, and the number of animals with pulmonary malignancies are given. The number of sacrificed animals and sacrificed animals with pulmonary malignancies are given in the last column. A l l times quoted in the following are to be understood as times from start of inhalation exposure.

In view of the difficult estimation of absorbed dose (20), the dose is given in conventional form as the product of concentration and exposure time. The unit working level month ( W L M ) is utilized. One working level is defined as a con­centration of radon daughters which results in the ultimate release of 1.3 X 105

MeV of a energy per liter of air; i t is equivalent to a concentration of 0.1 nCi radon (3.7 Bq) per liter of air in equilibrium with its four short-lived daughter products (7, 20). The time unit working month was defined as 170 hr. The unit W L M corresponds to 3.54 X 10~3 J m~ 3 hr.

The pathological classification followed a method described in earlier articles (27, 22). The pulmonary cancers were recorded as bronchogenic (type 1 in Table I ) or bronchoalveolar (type 2 in Table I ) according to histological criteria proposed by Masse (22). Epidermoid carcinomas and adenocarcinomas were, in view of the bronchic origin of their cells, given the joint pathological classification of bron­chogenic cancers. For cancers with cells of more peripheral origin a common clas­sification of bronchoalveolar carcinomas was used. Nontumorous histological changes were classified as adenomatosis, and benign tumors were classified as ad­enomas. Adenomatosis was very frequent in older animals, even in the nonexposed group; it is not considered in this study. For each cell type only the most severe lesion was recorded, even i f other lesions were present.

In the following analysis only malignant tumors, without regard to their patho­logical type, are considered. I t is felt that the distinction between adenomas and the malignant neoplasms is sufficiently well defined. In addition animals with ad­enomas only are a minor fraction (~15%) of all animals with neoplasms.

The spontaneous incidence of pulmonary malignancies in Sprague-Dawley rats is not well known. A review of data from various laboratories leads to an estimate between 10~3 and 5 X 10~3. The exact numerical value is not very critical for the present analysis; we have chosen the estimate of 3 X 10 - 3 . This agrees with the data of Sanders and Mahaffey (23) and is consistent with the observation of Mart in et al. (24) who found no pulmonary tumors in a life span observation of 485 Sprague-Dawley rats.

T A B L E I

Summary of Protocols of the Radon-Daughter Inhalation Experiments

Dose Daily Radon - No. of No. of No. of No. of sacrificed (Dose per duration No. of Total daughter No. of animals with animals with animals with animals (No. month) of inhalation inhalations duration concentration No. of animals pulmonary malignancy malignancy with pulmonary (WLM) (hr) per week (months) (WL) animals examined malignancy of type I of type 2 malignancy)

0 0 0 0 0 186 159 0 0 0 0 65 (50) 2 3 1.2 360 500 490 13 7 6 0

170 (110) 3 4 1.5 390 294 244 11 9 5 4 290 (45) 5 5 6 75 81 80 2 1 1 66 (2) 860 (1500) 5 5 0.6 2,500 20 17 4 2 2 0

1,470 (1500) 5 5 0.9 2,500 30 29 5 2 3 12 (0) 3,000 (1500) 5 5 1.9 2,500 40 39 17 4 13 2(1) 4,500 (1500) 5 5 2.9 2,500 49 48 25 15 14 11 (0) 9,250 (1500) 5 5 5.8 2,500 20 15 5 5 0 0 2,250 (4600) 16 4 0.5 3,000 25 21 7 4 3 0 3,700 (4600) 16 4 0.8 3,000 25 20 8 8 I 0 3,900 (1800) 6 4 2 3,000 50 48 16 15 1 0 5,400 (4600) 16 4 1.1 3,000 25 23 6 5 1 0 6,000 (2400) 7 5 2.3 3,000 35 35 9 6 4 0 7,000 (4600) 4 4 1.5 12,000 181 171 34 28 6 97 (12) 7,000 (7000) 5 5 0.9 12,000 33 14 3 3 0 0 8,000 (3000) 10 4 2.6 3,000 180 164 65 51 20 45 (22)

14,000 (4600) 4 4 2.9 12,000 79 72 1 1 0 12(1)

o X m r m <

m H

ESTIMATION OF THE PREVALENCE OF NEOPLASMS 593

An important question in the experimental studies of pulmonary neoplasms in animals is whether malignant neoplasms cause life shortening. In mammals with life spans much longer than the latency growth times of malignant neoplasms (primates, dogs) lethality is marked. There was no evidence of life shortening in our Sprague-Dawley rats, and there have been only a few cases where the death of an animal appeared to be related to a pulmonary tumor. As a rule, malignant neoplasms affect only limited parts of the lung tissue; furthermore, less than 1% of pulmonary malignancies were found to give rise to metastases.

Nevertheless, a pulmonary tumor could be lethal in different ways. An animal might be asphyxiated by a tumor of excessive size, or a tumor in a particular location might block the upper airways. In experiments where inhalation of plu-tonium was combined with benzo[a]pyrene given intratracheal^, large tumor sizes were frequently obtained (25) and cases of asphyxiation were ascertained upon autopsy. Similar tumor sizes were never reached in experiments with radon in­halation alone.

Finally, a tumor could cause massive hemorrhage or might secrete toxins. None of these various possibilities have been observed in the radon experiments. The failure to find any instance of blocked upper airways or of internal hemorrhage is consistent with the fact that most pulmonary tumors in the Sprague-Dawley rat are peripheral. Finally, no direct evidence exists for Sprague-Dawley rats, but experiments with grafts of pulmonary tumors to Wistar rats (26) have led us to the conclusion that the tumors do not secrete toxins that may contribute to mor­tality.

The experimental data do not indicate an increased prevalence of pulmonary malignancies in nonsacrified dead animals. However, a meaningful comparison is possible only in two groups where sufficient numbers of sacrifices and deaths from other causes occur within certain intervals. In the group of 181 animals exposed to 7000 W L M between Days 400 and 500 i t was found that 9 of 26 sacrificed animals had pulmonary malignancies; for the deaths from other causes this ratio was 5 to 27. In the group exposed to 8000 W L M after Day 600 there were 22 animals with malignancy among 45 sacrificed animals; the ratio for the other deaths was 22 to 41.

In view of these considerations, independence between pulmonary tumors and mortality appears to be an acceptable hypothesis for the purpose of the present study.

CONVENTIONAL ANALYSIS

This section gives numerical results obtained from a conventional analysis to facilitate the understanding of the subsequent section that deals with the more rigorous mortality-corrected analysis.

In the subsequent analysis only pulmonary malignancies—according to the cri­teria stated in the preceding section—are considered. The relevant information for each animal is then, apart from dose and duration of exposure, the time of its death and the presence or absence of a pulmonary malignancy. A more detailed treatment that discriminates between different types of pulmonary tumors is given in A p ­pendix A.

594 C H M E L E V S K Y ET AL.

800 | 1 1 1 \ 1 1 \ r

i zz 400 -

• high dose rotes

O low dose rotes

10 INHALATION DOSE / 10 WLM

-i r

F I G . 1. The mean lifetime, with standard error, in each group as a function of inhalation dose. Data from sacrificed animals are not included. The partition between low dose rates and high dose rates is set at 1600 WLM/month.

Figures 1-3 give results of a conventional analysis; the sacrificed animals (total number = 249) were excluded. Figure 1 represents, as a function of inhalation dose, the lifetime and its standard error in the individual groups; animals that could not be examined were also included. In this, as in subsequent figures, different symbols are used for the groups exposed to less than 1600 W L M / m o n t h and those exposed to more than 1600 W L M / m o n t h . This serves merely as a visual aid to distinguish results from "moderate dose rates" and "high dose rates." The distinction has no influence on the computations.

Figure 2 gives, separately for each group, the fraction of animals dead up to the specified time and the fraction of animals dead with a pulmonary malignancy.

Figure 3 gives the raw incidence of malignancies as a function of inhalation dose, i.e., the ratio of animals dead with pulmonary malignancy to the number of ex­amined animals in a group. These data are based only on animals that were ex­amined, and sacrificed animals were excluded since the temporal distribution of the sacrifices could seriously bias the results. The term "raw incidence" is used to emphasize the fact that this quantity is not corrected for life shortening. Without such a correction it is unclear whether the decline of the incidence at higher doses was exclusively due to life shortening, or whether there was also an inherent decline of the induction of pulmonary malignancies at high doses.

COMPETING-RISKS-CORRECTED ANALYSIS

The Problem of Fully Censored Data

To allow for competing risks the Kaplan-Meier estimate (4, 27) or similar es­timates (28) are appropriate in the analysis of lethal diseases. These methods are

E S T I M A T I O N O F T H E P R E V A L E N C E O F N E O P L A S M S 595

equally applicable i f a tumor is not lethal, but its time of appearance can be determined; an example is the study of mammary neoplasms (29). Under these conditions one speaks of right censored data; i.e., times to the tumor are either known or are known to exceed an observed time when an animal, still without tumor, has been lost for unrelated reasons. For nonlethal diseases that can be discovered only incidentally in a dead animal, the data are times, th of death with or without the disease present. The time to the effect is known to be smaller or

TIME /days TIME / days

F I G . 2. The fraction of animals dead without (single-shaded area) and with (double-shaded area) pulmonary malignancies as a function of time in the different experimental groups. These calculations are based on animals with pulmonary diagnostics; they do not include sacrificed animals.

596 C H M E L E V S K Y ET AL.

.6 -

ÜJ o z ÜJ 2 A

< tr

o low dose rates • high dose rates

INHALATION DOSE / I0 3 WLM

F I G . 3. Raw incidences as a function of inhalation dose. The raw incidence in a group is the ratio of the number of animals dead with pulmonary malignancy to the total number of animals. Sacrificed animals and animals without pulmonary diagnostics have been excluded. The standard errors are based on the binomial distribution. The partition between low and high dose rates is set at 1600 WLM/month.

larger than tt\ i t is never actually known. One may therefore speak of fully censored data? The mathematical analysis of such data involves the estimation of the prev­alence of the disease as a function of time. The prevalence at a given age can most readily be determined by serial killing experiments, but as pointed out by Hoel and Walburg (5), the analysis can also be performed from survival experiments, pro­vided the mortality rate is not influenced by the disease under study. Partially lethal diseases present a gray area where the estimation of prevalences from survival experiments is difficult or impossible.

The presence of a pulmonary tumor is determined upon necropsy in sacrificed animals, or in animals that died from other causes. Since independence of pul­monary tumors and of mortality is assumed, no distinction needs to be made between these two cases; dead animals, whether sacrificed or not, can be considered as random representatives of the animals at the specified time.

Before a discussion of the numerical methods, it is necessary to define the basic quantities which wil l be used. The definitions are in agreement with those in the report of the United Nations Scientific Committee on the Effects of Atomic Ra­diation (31). These definitions apply whether one deals with lethal or nonlethal diseases.

The tumor rate, r(t), is the probability per unit time of an animal to develop a tumor at time /; in mathematical statistics this or analogous functions are termed hazard functions.

The cumulative tumor rate, R(t), is the integral of the tumor rate up to time t:

3 The term doubly censored data (left and right censored) has also been used (30); however, the expression fully censored appears more appropriate to a situation where there are no exact observations.

E S T I M A T I O N O F T H E P R E V A L E N C E O F N E O P L A S M S 597

R(t)= P r(t')dt'. (1) Jo

The prevalence, P(t), is the probability that an animal bears a tumor at time t.

Of the three only P(t) can be obtained directly in the present case; however, R(t) or r(t) can be inferred indirectly. For this purpose one may express the prevalence in terms of the cumulative tumor rate, R(t):

P(t) = 1 - e x p ( - # ( 0 ) . (2)

R(t) is always larger than P(t), and it can exceed 1; at values of R(t) much smaller than 1 there is approximate numerical equality between P(t) and R(t).

Maximum Likelihood Estimate of the Prevalence

As stated, there are familiar methods for estimation of the prevalence in the case of right censored data (4, 5). These methods can be based on maximum likelihood considerations (52); however, they are sufficiently simple to be intuitively under­standable without consideration of their theoretical basis.

In contrast, the estimation of the prevalence for fully censored data is straight­forward only i f one restricts the analysis to one preselected time when all surviving animals are sacrificed and examined for pulmonary tumors. Ullr ich et al. (33) have used this approach in large-scale experiments that compare the prevalence or the mean number of pulmonary tumors in R F M / u n mice 9 months after exposure to X rays or fission neutrons. The simple solution can, in principle, also be applied when the analysis is aimed at the assessment of the time dependence of the tumor prevalence; however, i t requires serial sacrifice of animals so that there are enough necropsies in each preselected time interval. P(t) is then estimated as the observed fraction of tumor-bearing dead animals in a time interval. The limitation of this method is its costliness due to the large number of animals required. Experiments of this type [e.g., see (34)] are therefore rare. In the present study, 17 groups of animals were exposed to different doses with different durations of exposure. The total number of animals was roughly 1700; it is evident that a far greater number of animals would be required i f the prevalence were to be estimated separately for each group. However, it wi l l be seen that maximum likelihood methods permit estimates even from experiments with small groups of animals.

Let (t (i = 1, 2, . . . , / ) be the times of death of those animals that are found to carry a tumor and let tj (j' = 1, 2, . . . , / ) be the times of death of those animals that are found to be tumor free. The likelihood, A, is defined as the prob­ability for the actual outcome, conditional on the times of deaths and on the prev­alence function P(t):

A = n / > ( * + ) - n o - / > ( * / ) ) . (3) «=i j=\

For numerical calculations, i t is more convenient to use the log-likelihood:

In A = L = 2 In (/>(/+)) + 2 In (1 - P(tj)). (4) /=! 7=1

598 C H M E L E V S K Y ET AL.

65 WLM 490 ANIMALS

.4

UJ 0 o

Lü n

> ÜJ £ 4

- 170 WLM _

. 244 ANIMALS -

r "

290 WLM . 8 0 ANIMALS

1 1 r 4500 J • . 48

J •

1 —

2250 j -21 -

r

r 6000 .35

7000 . 14 -

300 600 900 0 300 600 900 0 300 600 900

TIME/DAYS

F I G . 4. Estimates of the prevalence of pulmonary malignancies from isotonic regression based on the combined data from life-span and sacrificed animals.

In Eqs. (3) and (4) i t is assumed that all times t? and tj are individually resolved. I f multiple deaths occur at a time, a more complicated relation, corresponding to the general binomial case, applies; it is given in Appendix B.

One can calculate the theoretical function P(t) that maximizes the log-likelihood. A condition for the solution is that the prevalence does not decrease in time. Wi th this constraint of monotonicity there exists a relatively simple algorithm to obtain P(t). The method, termed isotonic regression (55), has been applied by Hoel and Walburg (5) to radiation carcinogenesis studies.

Figure 4 gives the results of the isotonic regression for the 17 groups of the present experiment. In view of the small size of some of the groups it is not surprising that the shapes of the estimated curves vary widely. This type of analysis is therefore of limited value for a quantitative comparison and for the construction of dose-effect relations. A more efficient method is required that imposes additional con-

ESTIMATION OF THE P R E V A L E N C E OF NEOPLASMS 599

straints on the set of possible solutions. Such a method consists in postulating one underlying prevalence function (baseline function) that is assumed to vary with inhalation dose in a manner that depends on a chosen model. Inevitably the ut i­lization of a model wi l l introduce a degree of arbitrariness. To judge the bias that may be introduced in this way, three models wi l l be applied and compared that have been considered in the mathematical theory of survival data.

The maximum likelihood method wi l l be utilized in the present work in a modified form that leads to smooth curves. The monotonicity constraint is, for this reason, replaced by a convexity constraint (see Appendix B). An inconvenience of the convexity constraint is the requirement of a nonlinear optimization algorithm that takes the place of the simpler computation for isotonic regression; however, the more complex algorithm wi l l in any case be required in the simultaneous maximum likelihood analysis of several experimental groups.

SIMULTANEOUS MAXIMUM LIKELIHOOD ANALYSIS

Three Models for the Dose Dependence of the Prevalence

Of the three models which wi l l be considered, one—the shifted time model—has been suggested in studies of life shortening (36) and tumorigenesis (29, 37) after irradiation. The other two have been utilized in survival analysis and reliability studies (32) and in clinical and experimental investigations of cancer treatment (38, 39). The shifted time model postulates that the prevalence function, PD(t) is shifted toward earlier times in animals exposed to a dose D:

PD(t) = P(t + s(D)), (5)

where s(D) is the dose-dependent shift factor. The second distinct but related model assumes that the prevalence in the exposed

animals changes in a way that corresponds to an acceleration of time:

PD(t) = P(a(D)-t), (6)

where a(D) is the acceleration factor, also dependent on dose. In the statistical theory of survival analysis Eq. (6) is termed the accelerated failure time model (32).

The third, the proportional hazards model, postulates that the tumor rate, r(t), and therefore also the cumulative tumor rate, R(t), change by a dose-dependent factor \(D):

RD(t) = \(D).R(t). (7)

This model has been most extensively studied [see (32)] and is widely used with right censored data. Cox (40) has obtained a remarkably simple solution for the simultaneous maximum likelihood fit to such data; accordingly, this model is also referred to as Cox's model.

I t wil l be sufficient to consider the accelerated failure time model in detail; the corresponding equations for the other models are cited in Appendix B. The log-likelihood in the accelerated failure time model is

K lk Jk

L = 2 [ 2 In P(ak • ttk) + 2 In (1 - P{ak • tjk))], (8)

600 C H M E L E V S K Y ET AL.

where

ttk = time of death of animal i (i = 1, . . . , Ik) with malignancy in group k (k = 1, . . . , K\

tjk time of death of animal j (J' = 1, . . . , Jk) without malignancy in group k = ( * = l , . . . , / 0 ,

K - the number of groups. The numerical optimization procedure yields the baseline function P(f) and the! acceleration factors ak for the K groups. No analytical expression is postulated for; P(i). However, to ensure a smooth function convexity of In P(t) from above is used as a constraint.

Results of the Simultaneous Analysis \

For each of the three models, the simultaneous maximum likelihood fit has been performed with the 17 exposed groups. The control group was excluded because no pulmonary malignancy had been observed in this group. The optimization pro-; cedure utilizes the method of steepest descent in a general-purpose computer pro-! gram for nonlinear optimization developed by Abadie and Guigou. In an earlier publication dealing with the nonparametric fitting of dose-effect relations (41\ the characteristic features and requirements of such optimization programs have been discussed. Therefore only essential information relevant to the present case is given! in Appendix B.

For each of the three models, the maximum likelihood computation yields a best-fit prevalence function and a set of parameters [s(D), a(D), and X(D) in Eqs. (5) to (7)] for the individual groups. Figure 5 represents the results. I n each individual panel the histogram gives the fraction of animals that have died in the specified! time interval either without pulmonary malignancy (blank) or with malignancy! (shaded columns). The three curves on each panel give the estimated prevalences for the group according to the three models. These estimated prevalences are ex-j tended beyond the range of observed data in the individual panels. This is done for! a better indication of the characteristic differences among the three models, and! also because the inferred prevalence functions are utilized in the subsequent section for the construction of dose-effect relations.

The largest value of the combined likelihood corresponding to Eq. (8) ( L = -448.2) is obtained for the accelerated failure time model, but the values for! the shifted time model (L = -449.2) and the proportional hazards model (L = -448.6) are similar. From Fig. 5 one concludes that, in spite of characteristic differences of the prevalence functions and their variation with inhalation dose, these functions are similar for the relatively narrow time ranges with actual data in the individual groups. In the subsequent section it wi l l furthermore be found that incidences computed on the basis of the prevalence functions from the three models are similar, i.e., that the choice of the model is not critical for the resulting risk estimates. On the other hand, one would have to design experiments with a wider range of death times at each dose i f a discrimination between models were to be attempted. For moderate doses this might require more early sacrifices; for

E S T I M A T I O N O F T H E P R E V A L E N C E O F N E O P L A S M S 601

O

65 WLM (50 WLM/month) 490 ANIMALS

5400(4600) 23 J

7000(7000)

A

ju§fffnr 1 300 600 900

170 WLM (110WLM/mon 244 ANIMALS

rrrfft

h)

1470 r ~29

(1500)

/ > • "

y.y( n

3700 (4600) 20

6000(2400) ~35

n n aa fcJ l

300 600

TIME / DAYS

290WLM (35 WLM/month) "80 ANIMALS

3900(1800) 48

\

7000 (46 — 1 7 1

00)

/

140C 72

" 4 h

)0(46

m 00)

900 0 300 600 900

F I G . 5. The three estimates of the prevalence from the models. (Solid lines are from shifted time model; short broken lines are from accelerated time model; long broken lines are from proportional hazards model.) Each histogram gives the fraction of animals that died within each interval without (blank columns) and with (shaded columns) pulmonary malignancies. The data include sacrificed and life-span animals with pulmonary diagnostics.

high doses, where life shortening is substantial, large initial numbers of animals could be required with pulmonary diagnostics performed predominantly in animals that live relatively long.

Derivation of Mortality-Corrected Incidences

As stated earlier, the raw incidences do not represent a true dose dependence for the carcinogenic effect at dose levels where life shortening is substantial. I t is

602 C H M E L E V S K Y E T AL.

therefore desirable to compute a mortality-corrected quantity. One possibility is to define an adjusted incidence, I D , as the probability of the animals incurring a malignancy, provided the mortality rates were those of the controls. The formal statement of this definition is

m(t)-PD(t)dt, (9), 0

where m(t) is the fraction of the control population dying per unit time interval at time t. The value tmax = 850 days has been taken as the maximum survival time. In Fig. 6 the adjusted incidences are given for all three models. The incidences for the exposed groups are plotted in Fig. 6 as excess over the assumed small control incidence of 0.003.

The adjusted incidences from the accelerated failure time model and their stan­dard errors (see Appendix B) are compared to the raw incidences in Table I I . This table also contains the acceleration factors ak that correspond to the prevalence functions in Fig. 5. In the computations the baseline function, P(t), is defined only up to an arbitrary factor in time, and, conversely, the acceleration factors are defined only up to the same common factor. However, the normalization has been chosen so that P(t\ i.e., the assumed prevalence for the controls (a = 1), yields the incidence 0.003.

I t is of interest whether these results would fit a simple power function at low

8 Z ÜJ 9 o

CO 3

<

INHALATION DOSE / WLM

F I G . 6. Comparison of the adjusted incidences lD (minus control incidence of 3 X 10~ 3) as a function of the inhalation dose for the three basic models (ST: shifted time model; AT: accelerated time model; PH: proportional hazards model). The open symbols are used for the low-dose-rate range (<1600 W L M / month) and the solid symbols for the high-dose-rate range (>1600 WLM/month). Standard errors for all groups are given in Table II.

ESTIMATION OF T H E P R E V A L E N C E OF NEOPLASMS 603

T A B L E II

Comparison of Raw and Adjusted Incidences and Maximum Likelihood Estimates of Parameters for the Accelerated Failure Time Model

Dose D (dose per month) Adjusted Acceleration

(WLM) Raw incidence incidence fD factor ak

65 (50) 0.027 ± 0.007 0.025 (0.018-0.042) 1.30 (1.24-1.45) 170 (110) 0.046 ± 0.011 0.038 ±0.011 1.48 ± 0.10 290 (45) 0 (+0.12) 0.065 ± 0.06 1.68 ± 0.39 860 (1500) 0.24 ±0 .10 0.18 ± 0.07 2.20 ± 0.25

1,470 (1500) 0.29 ± 0.11 0.26 ± 0.07 2.45 ± 0.23 3,000 (1500) 0.43 ± 0.08 0.47 ± 0.08 3.25 ± 0.49 4,500 (1500) 0.68 ± 0.08 0.54 ± 0.05 3.75 ± 0.52 9,250 (1500) 0.33 ± 0.12 0.58 ± 0.07 4.2 ± 0.78 2,250 (4600) 0.33 ± 0.10 0.35 ± 0.09 2.73 ± 0.25 3,700 (4600) 0.40 ± 0.11 0.51 ± 0.10 3.51 ± 0.71 3,900 (1800) 0.33 ± 0.07 0.39 ± 0.06 2.86 ± 0.22 5,400 (4600) 0.26 ± 0.09 0.48 ± 0.07 3.18 ± 0.31 6,000 (2400) 0.26 ± 0.07 0.43 ± 0.07 3.03 ± 0.25 7,000 (4600) 0.30 ± 0.05 0.51 ± 0.04 3.50 ± 0.24 7,000 (7000) 0.21 ± 0.11 0.39 ± 0.08 2.85 ± 0.26 8,000 (3000) 0.36 ± 0.04 0.50 ± 0.05 3.45 ± 0.35

14,000 (4600) 0 (+0.03) 0.26 ± 0.08 2.45 ± 0.26

inhalation doses. However, this requires a somewhat arbitrary selection of a dose cutoff. A regression with the data below 1000 W L M results in the fit

ID = kDp, (10) with

k = 3.4 X 10~4 ± 1.4 X 10" 4 and p = 0.92 ± 0.07.

The value of k holds i f D is in the units W L M . I t is evident from this result, and in particular from the observation at 65 W L M ,

that a sublinearity at low doses cannot be excluded, even though linearity cannot be rejected on the basis of the present data. On the other hand, a power in dose substantially in excess of 1 can be rejected.

I f one assumes a linear dose dependence (p = 1), the linear regression yields the relation

I D = c-D with c = 2 . 0 X 1 0 " 4 ( ± 0 . 5 X 1 0 " 4 ) / W L M ( n )

= 0.056 ( ± 0 . 0 1 4 ) / ( J n T 3 hr).

I D is not the only meaningful quantity for the construction of a dose-effect relation. I t has been variously emphasized in studies of radiation risks (42, 43) that a realistic assessment of detriments would have to be based not on numerical incidences but on the loss of tumor-free life. Radiation-induced malignancies that appear late in life contribute less to the decrement than early malignancies. Ac­cordingly, it is of interest to compute a modified quantity, J D , that represents the

604 C H M E L E V S K Y ET AL.

average loss of tumor-free life span. Such a quantity that has been utilized in studies of mammary tumors in Sprague-Dawley rats (29, 37) has been termed effect period in the report of U N S C E A R (57). The definition of U N S C E A R does not include the mortality correction. Such a correction wi l l be applied, and the quantity effect period is then defined as

JD= fW M(t)-PD(t)dt, (12) Jo

where M(t) is the proportion of controls surviving at time /. Figure 7 gives the numerical values of JD. There is less agreement between the

different models for this quantity than for the adjusted incidences, I D .

CONCLUSIONS

The conventional analysis of the radon-daughter inhalation data permits the estimation of the dose dependence of life shortening and the dose dependence of the incidence of pulmonary tumors, uncorrected for life shortening. However, at higher doses the uncorrected incidence decreases due to life shortening. Maximum likelihood methods are required for the derivation of corrected incidences.

100

o o or CL­

IO

r T

• • S T

A a AT

<>• PH

1 r i 1 " l 0

. 9 , 1 8' i . o !

ö • •

-

0 A

•

0 6

0

A

A

• A •

0

A •

•

hi 1ALIGNANT NEOPLASMS

10 10 Z 1 0 3

INHALATION DOSE / WLM

10*

F I G . 7. The effect period JD (see text) as a function of inhalation dose for the three basic models. (ST: shifted time model; AT: accelerated time model; PH: proportional hazards model). The open symbols are used for the low-dose-rate range (<1600 WLM/month); the solid symbols for the high-dose-rate range (>1600 WLM/month).

ESTIMATION OF T H E PREVALENCE OF NEOPLASMS 605

A further limitation of the conventional analysis results from the impossibility I of mixing data from life-span experiments with data from sacrificed animals. In-I elusion or exclusion of the data from the sacrificed animals can, depending on the temporal distribution of sacrifices, have considerable influence on the results. For this reason, too, the analysis of the radon-daughter inhalation experiments was based on maximum likelihood methods.

The investigation of nonlethal diseases discovered incidentally at death results in fully censored data. As Hoel and Walburg have shown (5), isotonic regression must be utilized instead of the Kaplan-Meier estimate that is applicable to right censored data. Isotonic regression has been applied to estimate the survival-adjusted prevalence of pulmonary malignancies as a function of time after exposure in the various groups. The results of this isotonic regression reflect the general increase of the prevalence with dose and with time after exposure. But, due to the small size of most of the experimental groups, no overall estimate of the prevalence as a function of dose and time could be inferred. For this reason the simultaneous likelihood fit was required, and i t has provided the desired overall estimates. They are largely independent of the models chosen as the basis of the simultaneous fit. The same is true for the adjusted incidences obtained by integration over the prevalences.

The application of a simultaneous maximum likelihood fit is not new in itself. For right censored data a method based on the proportional hazards model has existed for some time and has been widely applied (32). No similar treatment had earlier been attempted with the fully censored data obtained for nonlethal diseases. As exemplified here with the radon studies, the computations require nonlinear optimization; such algorithms are widely available.

A n advantage of the method is the independence from any particular model. I t has therefore been applied not only with the proportional hazards models, but also with two other models that assume either a dose-dependent time shift of the prev­alence function or a change of the prevalence function that corresponds to an acceleration in time. A l l three models were nonparametric, i.e., no analytical expres­sion was postulated for the dose or the time dependence of the prevalence. The computational procedures are similar for the different models.

The methods that have been applied here to radon-daughter inhalation studies may also prove useful in other inhalation studies. However, they require that the pulmonary neoplasms do not appreciably contribute to mortality. While this appears to be the case with rats exposed to radon daughters, the condition may be not met in other species or with other radioisotopes.

APPENDIX A: SUMMARY OF RESULTS FOR DIFFERENT TYPES OF PULMONARY NEOPLASMS

Prevalences have been estimated in this article for pulmonary malignancies rather than all pulmonary tumors. The reason for this somewhat arbitrary restriction is the relatively small number of benign pulmonary tumors that have been found. I t was accordingly felt that the advantage of a sharper pathological criterion out­weighs the gain in statistical accuracy that would accrue from inclusion of the

606 C H M E L E V S K Y ET AL.

T 1 1 r

TIME/days

F I G . 8. Prevalance functions for the group exposed to 7000 WLM (4600 WLM/month) from the shifted time model for all pulmonary tumors (upper solid line), for pulmonary malignancies (lower solid line), for bronchogenic malignancies (broken line), and for bronchoalveolar malignancies (dotted line). |

benign tumors. There is nevertheless some interest in the discrimination between | benign and malign tumors and between malignancies from different cell types.

A separate analysis of benign tumors is not feasible because benign tumors have! been ascertained only in the absence of malignancies of the same cell type. In the! whole study, there are only 37 animals with benign tumors and no malignancies; there are, in addition, 4 animals where benign tumors have been found simulta­neously with malignancies in the other cell type. In contrast there were 231 animals with pulmonary malignancies. The number with malignancies of both cell types was 15.

In view of the relatively small numbers one cannot expect to establish, with statistical certainty, possible differences in the time course and dose dependence of the prevalence between malign and benign tumors. Nevertheless, Fig. 8 gives an overall comparison of the prevalence obtained for all pulmonary tumors (upper solid line), all pulmonary malignancies (lower solid line), bronchogenic malignan­cies (broken line), and bronchoalveolar malignancies (dotted line). The results are from a simultaneous fit of all groups, but Fig. 8 refers only to the large experimental group with 7000 W L M (4600 W L M / m o n t h ) . These additional computations have been performed with the shifted time model; similar results were obtained with the other models.

The derived time shifts for the different groups and for the different types of neoplasms are not given in tabular form but in correlation diagrams in Fig. 9.

ESTIMATION OF T H E PREVALENCE OF NEOPLASMS 607

* 800

o

\ 6 0 0

- L00 x

LU 200 2

all pulmonary tumors

bronchogenic malignancies

bronchoalveolar malignancies

0 2 0 0 L00 6 0 0 0 2 0 0 £00 6 0 0 0 200 400 600

T I M E S H I F T / d a y s (all malignancies)

FIG. 9. Correlation diagrams of the time shifts for all pulmonary malignancies (abscissa) versus those for all pulmonary tumors, for bronchogenic malignancies, and for bronchoalveolar malignancies.

Among all pulmonary malignancies, all pulmonary tumors, and the bronchogenic malignancies no systematic difference is apparent. However, one notes that the time shifts for bronchoalveolar malignancies at high doses tend to be less than those for the other neoplasms.

Figure 10 compares the adjusted incidences for the different types of pulmonary tumors. A decrease at high doses and higher dose rates is apparent only for the bronchoalveolar malignancies. This is in line with Fig. 9.

APPENDIX B: ESSENTIALS OF THE NUMERICAL COMPUTATIONS

Likelihood Equation for Finite Time Intervals

Equation (4) for the log-likelihood contains the exact times of death of the animals. The actual computations are performed on a discrete time grid with in­tervals of 30 days or, in the case of the accelerated time model, with a logarithmic interval of width 0.08. The data are related to this grid by suitable interpolation. In this way the total number of variables to be optimized in a computation is reduced to the estimates at 50 grid points for the prevalence function and to 16 additional variables for the parameters for the individual groups. The log-likelihood is then

/

1= 2 [Mi In (Pit,)) + (N, - In (1 - Pit,))}, ( B l ) i=l

where tt (/ = 1 to / ) are the discrete times, TV, is the total number of deaths in interval i , and M , is the number of animals with malignancy in interval i.

The simultaneous solution for several experiments requires, even for the simple constraint of monotonicity, an iterative optimization algorithm; the method of steep­est descent in a computer version, GRGA, of Abadie and Guigou 4, has been applied in the present analysis. Wi th the GRGA algorithm it is readily possible to replace

4 J. Abadie and J. Guigou, Gradient reduit generalise. Internal Report HI-069/02, Electricite de France, 1969.

608 C H M E L E V S K Y ET AL.

LU O

O

bronchoalveolar malignancies

0 £ _

Q .2 ÜJ

3 ofc_

2 -er <

° bronchogenic malignancies

Ol

all pulmonary tumors

low dose ra tes high dose rates

8 10 12 14

I N H A L A T I O N D O S E / I C T W L M

F I G . 10. Comparison of the adjusted incidences for all pulmonary malignancies, for bronchogenic malignancies, and for bronchoalveolar malignancies. Open dots are for the low-dose-rate range (<1600 WLM/month); solid dots for the high-dose-rate range (>1600 WLM/month).

the monotonicity constraint by a constraint that leads to smooth curves. The con­straint actually used was therefore convexity of In (P(t)). Figure 11 uses the example of a single but relatively large group for a comparison of the naive estimates (dots), the isotonic regression with the exact times of deaths (step function), and the computation with convexity constraints (continuous curve) applied only to this individual group.

Joint Likelihood for the Shifted Time Model and the Proportional Hazards Model

From Eqs. (4) and (5) one obtains the following relation for the log-likelihood in the shifted time model:

L = 2 [2 In (P(ttk + sk) + 2 In (1 - P(tjk + sk))] (B2)

ESTIMATION OF T H E P R E V A L E N C E OF NEOPLASMS 609

40 f 1 1 1 1 1 r

O (/>

TIME/days

F I G . 11. Illustration of the results of maximum likelihood estimation of the prevalence by its application to a single large group of animals exposed to an inhalation dose of 7000 WLM. The upper panel represents the total number of animals which died in the intervals (blank areas: animals without pul­monary malignancies; shaded areas: animals with pulmonary malignancies). The lower panel gives the estimated prevalence functions (step function from isotonic regression; continuous function from regres­sion with convexity constraint). Dots are the ratios of the number of animals dead with malignancies to the total number of animals dead.

[symbols as in Eq. (8)] . The proportional hazards model leads to a more complicated relation because

the baseline function in this model is not the prevalence but the related quantity cumulative tumor rate, R(t). Wi th the proportional hazards assumption [Eq. (7)] and the relation [see Eq. (2)]

one obtains

and

R(t) = - I n (1 - P(t)) (B3)

PD(t) = 1 - exp( - X(Z)) • R(t)) (B4)

L = 2 [2 In (1 - exp(-A*.tf(4))) " 2 A * ( B 5 ) k=\ /=! y=l

The same computer program is used for both the shifted time model and the accelerated time model; this requires the utilization of the logarithm of time in the case of the accelerated time model. Wi th the monotonicity constraint, convergence of the optimization procedure is fast and insensitive to the initial estimates. Wi th the convexity constraint convergence can be problematic, and the proper selection

610 CHMELEVSKY ET AL.

ST

600 800 1000 1200 1400 1600 TIME + SHIFT / days

F I G . 12. The prevalence from the shifted time model as a function of converted time (actual time plus the time shift that has been obtained for each group relative to the control group). The upper panel represents the pooled total of animals from all groups that died without (blank columns) and with (shaded columns) pulmonary malignancies in each interval. Dots are the ratios of the number of animals dead with pulmonary malignancies to the total number of animals dead.

of initial estimates is critical. The initial estimates were usually obtained on the basis of the results of the simpler optimization with monotonicity as a constraint. Repeated computations with varying starting points have been performed to verify the stability of the solutions.

The maximum likelihood values that were obtained, overall and for the individual groups, were within the theoretically expected range of values. Presentation of the full data is impractical, but Figs. 12 and 13 are included to illustrate the quality of the overall fit. In these figures the individual groups of animals are superimposed according to the respective maximum likelihood shifts or acceleration factors. The resulting histograms of the total number of animals and the number of animals with malignancies are given in the upper panels of the graphs. In the lower panels the maximum likelihood prevalence is compared to the ratios of animals with malignancy to all animals in the respective intervals. These curves help in judging the applicability of the models.

The standard errors for the raw incidences in Fig. 3 were obtained by using the binomial distribution (for 0 observed events the 75% confidence range was used). The standard errors for the adjusted incidences were obtained through the standard

ESTIMATION OF T H E PREVALENCE OF NEOPLASMS 611

120

CO _i < I 8 0 -z < o

0 1000 2000 3000 TIME x ACCELERATION FACTOR / days

F I G . 13. The prevalence from the accelerated time model as a function of converted time (time multiplied by the acceleration factor that has been obtained for each group relative to the control group). The upper panel represents the pooled total of animals from all groups that died without (blank columns) and with (shaded columns) pulmonary malignancies. The missing part of the histogram represents 14% of all animals included in the computation. Dots are the ratios of the number of animals dead with pulmonary malignancies to the total number of animals dead.

errors of the acceleration factors or through the corresponding parameters for the other models. The standard errors of the parameters are obtained on the basis of the Fisher statistics as described in a previous application of the optimization procedure (41). This is an approximate method, applicable to large numbers of animals. The errors are not symmetrical but have been averaged in Table I I except for the first group where the asymmetry is particularly large. The parameters and their standard errors are given only for the accelerated time model.

ACKNOWLEDGMENTS

We thank our colleagues Benno Schüllner and Stepänka Riedmann for valuable help in the compu­tations. Critical comments and useful advice from Drs. Harald H. Rossi and Art Petersen were greatly appreciated.

AT

R E C E I V E D : September 10, 1981; R E V I S E D : February 1, 1982

612 C H M E L E V S K Y ET AL.

R E F E R E N C E S

/. ICRP, Biological Effects of Inhaled Radionuclides. Report No. 31, International Commission on Radiological Protection, Pergamon, Oxford/New York/Frankfurt, 1980.

2. National Research Council, Committee on the Biological Effects of Ionizing Radiations, The Effects on Populations of Exposure to Low Levels of Ionizing Radiation. Natl. Acad. Sei., Washington, DC, 1980.

3. United States Radiation Policy Council, Report on the Task Force on Radon Structure. RPC-80-002, Washington, DC, 1980.

4. E. L. KAPLAN and P. M E I E R , Non-parametric estimation from incomplete observations. J. Am. Stat. Assoc. 53, 457-481 (1957).

5. D. G. H O E L and H. E. WALBURG, Statistical analysis of survival experiments. J. Natl. Cancer Inst. 49, 361-372 (1972).

6. J. LAFUMA, Les radioelements inhales. In Radioprotection, Vol. 9, pp. 15-25. Dunod, Paris, 1974. 7. J. LAFUMA, Cancers pulmonaires induits par differents emetteurs alpha inhales. Evaluation de

1'influence de divers parametres et comparaison avec les donnees obtenues chez les mineurs d'uranium. In Late Biological Effects of Ionizing Radiation, Vol. II, pp. 531-541. International Atomic Energy Agency, Vienna, 1978. [STI/PUB/489.]

8. J . LAFUMA, J . CHAMEAUD, R. PERRAUD, R. MASSE, J . C. NENOT, and M. M O R I N , Etude exper-imentale de la comparaison de Taction toxique sur les poumons de rat des emetteurs a de la serie des actinides. In Radiation Protection in Mining and Milling of Uranium and Thorium, pp. 43-53. Labour Office, Geneve International, 1976.

9. J. CHAMEAUD, R. PERRAUD, J. LAFUMA, R. MASSE, and J. PRADEL, Lesions and lung cancers induced in rats by inhaled radon 222 at various equilibrium with radon daughters. In Experimental Lung Cancer (E. Karbe and J. F. Park, Eds.), Vol. 1, pp. 411-421. Springer-Verlag, New York, 1974.

10. J. CHAMEAUD, R. PERRAUD, R. MASSE, and J. LAFUMA, Cancers induced by 222 radon in the rat. In The Assessment of Radon and Daughter Exposures and Related Biological Effects, in press.

/ / . M. MORIN, J . C. NENOT, R. MASSE, D. NOLIBE, H. METIVIER, and J . LAFUMA, Induction de cancers chez le rat apres inhalation de radioelements emetteurs alpha. In Biological and Envi­ronmental Effects of Low-Level Radiation, Vol. II. pp. 109-119. International Atomic Energy Agency, Vienna, 1976. [STI/PUB/409.]

12. J. LAFUMA, J. C. NENOT, M. MORIN, R. MASSE, H. METIVIER, D. NOLIBE, and W. SKUPINSKI, Respiratory carcinogenesis in rats after inhalation of radioactive aerosols of actinides and lan-thanides in various physicochemical forms. In Experimental Lung Cancer (E. Karbe and J. F. Park, Eds.), pp. 443-453. Springer-Verlag, Berlin/Heidelberg/New York, 1974.

13. D. NOLIBE, M. DISCOUR, R. MASSE, and J. LAFUMA, The effects of immune modulation on plu-tonium dioxide lung carcinogenesis in the rat. In Pulmonary Toxicology of Respirable Particles (C. L. Sanders, F. T. Cross, G. E. Dagle, and J. A. Mahaffey, Eds.), pp. 536-540. USDOE Technical Information Center, Oak Ridge, TN 1980. [Available as CONF-791002 from National Technical Information Service, Springfield, VA 22161].

14. R. PERRAUD, J . CHAMEAUD, R. MASSE, and J . LAFUMA, Cancers pulmonaires experimentaux chez le rat apres inhalation de radon associe ä des poussieres non radioactives. C. R. Acad. Sei. Ser. D 270, 2594-2595 (1970).

15. J. CHAMEAUD, R. PERRAUD, J. C H R E T I E N , R. MASSE, and J. LAFUMA, Etude experimentale de Taction combinee de la fumee de cigarettes et du depot actif du radon-222. In Late Biological Effects of Ionizing Radiation, Vol. 11, pp. 429-436. International Atomic Energy Agency, Vienna, 1978. [STI/PUB/489.]

16. J. CHAMEAUD, R. PERRAUD, J. C H R E T I E N , R. MASSE, and J. LAFUMA, Combined effects of in­halation of radon daughter products and tobacco smoke. In Pulmonary Toxicology of Respirable Particles (C. L. Sanders, F. T. Cross, G. E. Dagle, and J. A. Mahaffey, Eds.), pp. 551-557. USDOE Technical Information Center, Oak Ridge, TN, 1980. [Available as CONF-791002 from National Technical Information Service, Springfield, VA 22161.]

17. J. CHAMEAUD, R. PERRAUD, J. LAFUMA, and R. MASSE, Cancers du poumon experimentaux provoques chez le rat par des inhalations de radon C. R. Acad. Sei. Ser. D 273, 2388-2389 (1971).

ESTIMATION OF T H E PREVALENCE OF NEOPLASMS 613

18. R. PERRAUD, J. CHAMEAUD, J. LAFUMA, R. MASSE, and J. C H R E T I E N , Cancer bronchopulmonaire experimental du rat par inhalation de radon. Comparaison avec les aspects histologiques des cancers humains. J. Fr. Med. Chir. Thorac. 26(1), 25-41 (1972).

19. J. CHAMEAUD, R. PERRAUD, R. MASSE, J. C. NENOT, and J. LAFUMA, Cancer du poumon provoques chez le rat per le radon et ses descendants ä diverses concentrations. Comparaison de la relation dose-effect chez l'homme et chez Tanimal. In Biological and Environmental Effects of Low-Level Radiation, Vol. II, pp. 223-338. International Atomic Energy Agency, Vienna, 1976. [STI/PUB/ 409.]

20. W. JACOBI, Relations between the inhaled potential alpha-energy of 2 2 2 Rn- and 220Rn-daughters and the absorbed alpha-energy in the bronchial and pulmonary region. Health Phys. 23, 3-11 (1972).

21. J. L A F U M A and R. MASSE, Etude experimentale des polluants radioactifs inhales. In Les collogues de VINSERM, Vol. 29, pp. 307-324. Institut National de la Sante et de la Recherche Medicale, Paris, 1974.

22. R. MASSE, Histogenesis of lung tumors induced in rats by inhalation of a emitters: An overview. In Pulmonary Toxicology of Respirable Particles (C. L . Sanders, F. T. Cross, G. E. Dagle, and J. A. Mahaffey, Eds.). USDOE Technical Information Center, Oak Ridge, TN, 1980. [Available as CONF-791002 from National Technical Information Service, Springfield, VA 22161.]

23. C. L . SANDERS and J. A. M A H A F F E Y , Toxicology of transuranium in rodents. In Biological Im­plications of Radionuclides Released from Nuclear Industries, Vol. 1, pp. 89-104. International Atomic Energy Agency, Vienna, 1979.

24. J. C. M A R T I N , H. DANIEL, and L . L E BOUFFANT, Short- and long-term experimental study of the toxicity of coal-mine dust and of some of its constituents. In Inhaled Particles IV (W. H. Walton, Ed.), pp. 361-371. Pergamon, Oxford/New York, 1977.

25. H. M E T I V I E R , R. MASSE, I. L ' H U I L L I E R , and J. LAFUMA, Etude de Taction combinee de 1'oxyde de plutonium inhale et de deux cancerogenes chimiques de l'environnement. In Biological Im­plications of Radionuclides Released from Nuclear Industries, Vol. II, pp. 93-103. International Atomic Energy Agency, Vienna, 1979.

26. D. N O L I B E , R. MASSE, P. FRITSCH, and J. LAFUMA, Greffe chez la souris athymique de tissu pulmonaire de rat contamine par 1'oxyde de plutonium. In Biological and Environmental Effects of Low-Level Radiation, Vol. II, pp. 95-107. International Atomic Energy Agency, Vienna, 1976. [STI/PUB/409.]

27. R. PETO, Guidelines on the analysis of tumour rates and death rates in experimental animals. Br. J. Cancer, 29, 101-105 (1974).

28. W. N E L S O N , Theory and applications of hazard plotting for censored failure data. Technometrics 14, 945-966 (1972).

29. C. J. SHELLABARGER, D . C H M E L E V S K Y , and A. M. K E L L E R E R , Induction of mammary neoplasms in the Sprague-Dawley rat by 430-keV neutrons and X-rays. J. Natl. Cancer Inst. 64, 821-833 (1980).

30. B. W. T U R N B U L L , Nonparametric estimation of a survivorship function with doubly censored data. J. Am. Stat. Assoc. 69, 169-173 (1974).

31. United Nations Scientific Committee on the Effects of Atomic Radiation, Sources and Effects of Ionizing Radiation. United Nations, New York, 1977.

32. J. D . K A L B F L E I S C H and R. L . PRENTICE, The Statistical Analysis of Failure Time Data. Wiley, New York, 1980.

33. R. L . U L L R I C H , M. C. JERNIGAN, and L . M. ADAMS, Induction of lung tumors in RFM mice after localized exposures to X rays or neutrons. Radiat. Res. 80, 464-473 (1979).

34. R. J. M. F R Y , E. STAFFELDT, and S. A. T Y L E R , Some problems arising in analysis of large-scale animal irradiation experiments. Environ. Int. 1, 361-366 (1978).

35. R. E. BARLOW, D. J. BARTHOLOMEW, J. M. BREMNER, and H. D . BRUNK, Statistical Inference under Order Restrictions; The Theory and Application of Isotonic Regression. Wiley, New York, 1972.

36. G. W. B E E B E , C. E. LAND, and H. KATO, The hypothesis of radiation-accelerated aging and the mortality of Japanese A-bomb victims. In Late Biological Effects of Ionizing Radiation, Vol. 1, pp. 3-27. International Atomic Energy Agency, Vienna, 1978. [STI/PUB/489.]

37. C. J. SHELLABARGER, R. D . BROWN, A. R. RAO, J. P. SHANLEY, V. P., BOND, A. M. K E L L E R E R , H. H. ROSSI, L . J. GOODMAN, and R. E. M I L L S , Rat mammary carcinogenesis following neutron

614 C H M E L E V S K Y ET AL.

or x-radiation. In Biological Effects of Neutron Irradiation, pp. 391-401. International Atomic Energy Agency, Vienna, 1974.

38. R. PETO, M. C. P I K E , P. ARMITAGE, N. E. BRESLOW, D. R. C O X , S. V. HOWARD, N. M A N T E L , K . MCPHERSON, J. PETO, and P. G . SMITH, Design and analysis of randomized clinical trials requiring prolonged observations of each patient. II. Analysis and examples. Br. J. Cancer 35, 1-39 (1977).

39. R. L. PRENTICE and L. A. G L O E C K L E R , Regression analysis of grouped survival data with application to breast cancer data. Biometrics 34, 57-67 (1978).

40. D. R. Cox, Partial likelihood. Biometrika 62, 269-276 (1975). 4L A. M. K E L L E R E R , D. C H M E L E V S K Y , and E. J. H A L L , Nonparametric representation of dose-effect

relations. Radiat. Res. 84, 173-188 (1980). 42. H. G . PARETZKE, Late effects of low doses and dose rates. In Advances in Radiation Dosimetry

and Protection in Medicine (V. Perez-Mender and R. H. Thomas, Eds.), pp. 97-114. Plenum, New York, 1980.

43. D. CHMELEVSKY, Radiation carcinogenesis at low doses of different radiations—Consideration of some recent findings. In Proceedings, 7th Symposium on Microdosimetry (J. Booz, H. G . Ebert, and H. D. Hartfiel, Eds.), Vol. 1, pp. 49-63. Commission of the European Communities, Harwood, London, 1981.